As @kelvin has mentioned in a commentary, one of the members of Vercel has explicitly stated:
Hi, this is expected. Next.js is not relying on dependencies vs. devDependencies either if you host on platforms like Vercel or use output: “standalone”.
So the proper answers to your questions are:
Is Next.js 13 do the minification itself for me?
Yes it will minify for you and it seems that it doesn’t rely on devDependencies in order to achieve it.
Should I separate them manually, defying create-next-app’s recommendation?
No, according to Balázs Orbán you shouldn’t or, at least, you don’t have to. But I think that as it is not required by next.js it is not forbidden also, because many of the official examples are still separating them.
Please note that, for react applications that don’t use next.js (and also other kind of applications based on npm and yarn package managers) the separation between dependencies and devDependencies is still necessary.
Note: Kelvin has already answered it. But as a commentary it was a little bit hidden. I’m posting it as an answer with some more explanation just to highlight the answers for those, like me, that get here by searching.